$s$-harmonic functions in the small order limit

We study families $u_s$ of functions satisfying the equations $(-\Delta)^s u_s=0$, $s \in (0,1)$ in a smooth bounded open set $\Omega \subset \mathbb{R}^N$. The main purpose of this paper is twofold. First, we provide a detailed analysis of the asymptotics of these families in the zero order limit $s \to 0^+$. Second, we study the differentiability of $u_s$ as a function of $s$. Most of our results are devoted to the associated Poisson problem, where the family $u_s$ is determined by the exterior condition $u_s = g$ in $\mathbb{R}^N \setminus \Omega$ for some fixed function $g \in L^\infty(\mathbb{R}^N \setminus \Omega)$. Our results show that both the zero order asymptotics and the differentiability properties of $u_s$ can be expressed in terms of the logarithmic Laplacian of suitable extensions of $g$. This allows to deduce pointwise monotonicity properties of $u_s$ in the order parameter $s$ for a large class of functions $g$.